All pieces are made of cheap plastic, but can't be bent.
I can insert the white stick into the smaller gear and spin it like a top. I can also spin the bigger gear, with my hands, with the knobby thing on the bottom side. Don't know how they all fit together though. And I can't seem to find any use for the smaller, green stick piece.

It may be of academic interest to calculate the probability of winning, and we can discuss the semantics of what the exact phrasing implied, but I cannot see any possible practical reason to change your betting strategy to improve chances of a win rather than maximising expected winnings.

You're considering a win vs no win scenario. What you also need to take into account is the non-zero probability of drawing multiple winners from tickets drawn from different rolls, which give you a higher return. If you get 2 winners, you win double the amount, which is impossible to get from the same roll (assuming a roll has 1 winner).
You can calculate the expected return using the binomial as well, but you need to multiply the probability with the reward too.

Um, nope. Think of it this way: Say there is a winner in every consecutive set of 50. You buy ten of these. What are the chances you got the winner on your hands? 10/50, clearly.

Now consider when you buy 10, so that each one is from a different set of 50. What are the chances the first one is a winner? 1/50. The second one? Same, 1/50. Each of them has a 1/50 chance of being a winner. So in expectation, you have 10*(1/50) = 10/50 winners - which is exactly the same as the first case.

EDIT: More.

But if you know that all of the 15 that were bought just before you in the roll turned out to not have any winners. Now if you buy the next ten tickets, your chances of picking a winner increase to 10/35.

"Then, there is a horrifying 512-ounce version that they call child size. How is this a child-sized soda?"
"Well, it's roughly the size of a two-year old child, if the child were liquefied. It's a real bargain at $1.59."http://i.imgur.com/WvUJwio.jpg

I'm running it on a free AWS EC2 instance which takes way too long to run any image wider than 1024 pixels. I'll try to scale it up though. I agree, this would look much more magnificent at the original resolution.